I’m not familiar with mixed effects. Ny naive approach to this would probably involve using a Kalman filter or equivalent to extract the two ‘unobserved’ components at a given site. I’ll have a bit more of a think…

1) An older method common in the 1980s was to fit a cubic spline to each core; obtain the deltas from the fit and average the deltas. Dendrochronologists in the 1990s observed that this removed any long-term information (although series formed in this way were used in the programmatic article Hughes and Diaz 1994, still cited, to criticize earlier work which did not rely on tree rings.)

2) The individual cores, viewed s time series, are heavily autocorrelated. Some dendrochronologists did ARMA fitting to individual cores (prewhitening), obtained the deltas from that and averaged them. Removing the autocorrelation in the cores tends to leave a chronology with little long-term variation (e.g. Stahle chronologies used in MBH98).

3) Some chronologists (e.g. Jacoby, Graybill) fitted each core with a generalized negative exponential (neg exp plus a constant) or, if this didn’t fit, a horizontal or negative-trending straight line. This is option 2 in COFECHA. It’s usually what is meant by “conservative standardization”. There are some curious numerical issues in how this is implemented in COFECHA, but they don’t have a big impact. Cook and Peters 1997 criticized this method as potentially producing biased end effects – their examples ironically were a bristlecone (Campito Mt) and Gaspe.

Virtually every chronology archived at WDCP as a *.crn series is one of the above 3 types.

4) The term “RCS” standardization describes a procedure which has become popular in the 1990s – calculating a generalized negative exponential fit for the entire site, and calculating the deltas for each core from the one fit. This results in more centennial scale variability. In some of the sites of interest to Briffa (e.g. Polar Urals), the cores are very short on average (median – 150 years) and so standardizing individual cores eliminates any changes in the population mean.

I think that there are some interesting statistical issues involved in dendro standardization. A couple of years ago – is it that long that I’ve been doing this – I experimented with doing ring width standardization using mixed effects methods and got some really excellent results. The advantage of this is that I was able to put some of these calculations in a more general statistical framework and show how the little ad hoc dendro recipes fit into this framework and how to use the diagnostics available from the mixed effects modeling for dendro purposes.

In mixed effects terms (say the nlme package of Pinheiro and Bates), “conservative” standardization is simply a type of nlsList fit; while RCS standardization is an nls-fit. I can do either chronology in a couple of lines of code and replicate standard results applying a more general statistical method. The dialectic between nlsList and nls modeling (or lmList and lm modeling) observed in mixed effects texts is exactly and I mean EXACTLY identical to the tension between RCS and conservative standardization in dendro.

In mixed effects terms, you have interesting issues: there is an individual effect as mean growth rates vary by tree; there is an aging effect; there is a random effect from annual climate (reserving temperature/precipitation issues); you have autocorrelation in the residuals; you’ve got every issue contemplated in mixed effects and then some.

In addition, the dendro data sets are large and rich. While the mixed effects software in 2004 could deal with combinations of these issues, it didn’t quite deal with all of them. However, you could get a lot done with what was there. It would be an interesting project to follow up on this. It’s a big project; it would work like a champ; but I’ve got overtaken with too many other things. It would be a great project for a grad statistics student.

However, since this issue has come up in the context of a “practical” issue, maybe I’ll re-visit it. So much to do, so little time.

I’d really like to understand.

Perhaps the first step is for me to understand the processing you do to the raw series better. I understand that it is to adjust for variable growth by age of the tree – anything else? Can you direct me to any references that would be accessible via the Internet? I’m also missing something on this detrending process – aren’t you ultimately looking for a series with a ‘trend’ i.e. the signal of interest? Is the concern with heteroskedasticity really a concern about trends rather than variability per se?

Which is an entirely different level of ‘non-linear’ (you can’t even make first order Taylor expansion/log-linearisation arguments with this one). It requires temperature to be the limiting factor throughout the entire (or near enough) reconstruction history for them to be valid.

RW = A1*X*Y*(….)

because if the soil is infertile the tree dies, if it’s too dry the tree dies,… It’s not possible to get that kind of result from a simple linear relationship.

This whole episode illustrates the trouble people get into when doing a statistical analysis. Statistics isn’t magic. Statistics can’t extract information that isn’t there. Unless there is a prior understanding of the mathematical relation among all the variables affecting a proxy and independent control or measurement of the uninteresting variables, there is no way to honestly extract the variable of interest, temperature in this case. With all the knowledgeable people contributing to this blog, I have yet to see any indication that the relation between tree rings/density and tree growth factors(temperature, water, sunlight,fertility,…) is understood in any quantitative way. Wilson’s use of low variance is unjustified unless it can be shown, a priori, that it somehow keeps the uninteresting variables constant. Absent that, it’s just a form of cherry picking applied after the fact to get time series that have the desired correlation with temperature. In this context, both the Polar Urals and Yamal time series are irrelevant as thermometers since we don’t know all the conditions that controlled the tree growth.